US5923770A - 3D cardiac motion recovery system using tagged MR images - Google Patents
3D cardiac motion recovery system using tagged MR images Download PDFInfo
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T7/00—Image analysis
- G06T7/20—Analysis of motion
- G06T7/246—Analysis of motion using feature-based methods, e.g. the tracking of corners or segments
- G06T7/251—Analysis of motion using feature-based methods, e.g. the tracking of corners or segments involving models
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06V—IMAGE OR VIDEO RECOGNITION OR UNDERSTANDING
- G06V10/00—Arrangements for image or video recognition or understanding
- G06V10/70—Arrangements for image or video recognition or understanding using pattern recognition or machine learning
- G06V10/74—Image or video pattern matching; Proximity measures in feature spaces
- G06V10/75—Organisation of the matching processes, e.g. simultaneous or sequential comparisons of image or video features; Coarse-fine approaches, e.g. multi-scale approaches; using context analysis; Selection of dictionaries
- G06V10/754—Organisation of the matching processes, e.g. simultaneous or sequential comparisons of image or video features; Coarse-fine approaches, e.g. multi-scale approaches; using context analysis; Selection of dictionaries involving a deformation of the sample pattern or of the reference pattern; Elastic matching
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T2207/00—Indexing scheme for image analysis or image enhancement
- G06T2207/10—Image acquisition modality
- G06T2207/10072—Tomographic images
- G06T2207/10088—Magnetic resonance imaging [MRI]
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T2207/00—Indexing scheme for image analysis or image enhancement
- G06T2207/30—Subject of image; Context of image processing
- G06T2207/30004—Biomedical image processing
- G06T2207/30048—Heart; Cardiac
Definitions
- the present invention relates to the recovery of 3D cardiac motion from a volunteer dataset of tagged MR images and more specifically to a system that includes global models with parametric offsets, constant volume constraints for cardiac motion recovery and tessellation of the model.
- Terzopoulos and Metaxas included a global superquadric component in their deformable model.
- the deformations from this base superquadric model take( the form of a thin membrane spline described using the Finite Element Method (FEM).
- FEM Finite Element Method
- the default shape of the model is a thick ellipsoid rather than a shape closer to a real LV.
- their model does not provide a concise description of the LV movement. Rather, piecewise plots describe the motion.
- their model assumes a dense tag acquisition and therefore makes no use of "regularizing" constraints.
- the present invention includes an overall model, a geodesic-like prismoidal tessellation of the model and constant volume constraints.
- the overall model is a new solid shape model formulation that includes built-in offsets from a base global component (e.g. an ellipsoid) which are functions of the global component's parameters.
- the offsets provide two features. First, they help to form an expected model shape which facilitates appropriate model data correspondences. Second, they scale with the base global model to maintain the expected shape even in the presence of large global deformations.
- the geodesic-like prismoidal tessellation of the model provides for more stable fits.
- the constant volume constraints are imposed to infer the motion of the left ventricle where the tag intersections are sparsely distributed.
- the present invention is applied to the recovery of 3-D cardiac motion from a volunteer dataset of tagged-MR images.
- FIG. 1 illustrates a block diagram of the present invention.
- FIGS. 2a, 2b and 2c illustrate fitting a single model to data from two orthogonal tag acquisitions in order to recover an estimate of 3-D LV motion.
- FIGS. 3a and 3b illustrate model formulation which is made up of three components.
- the base global model and parametric offsets are shown in FIG. 3a and local deformations forming the overall model are shown in FIG. 3b.
- FIG. 4 illustrates calculation of parametric, offset vectors.
- FIGS. 5a, 5b and 5c illustrate a spherical model with offsets, a radially scaled model with parametric offsets and a radially scaled model with non-parametric offsets respectively.
- FIGS. 6a, 6b and 6c illustrate an original model, a globally twisted model with parametric offsets and a globally twisted model with non-parametric offsets respectively.
- FIGS. 7a and 7b illustrate a comparison of different model tessellations.
- FIG. 8 illustrates a model breaking under high stress during recovery.
- FIGS. 9a and 9b illustrate that under both the minimal stretching and constant volume constraints, the sealed tube tessellation is biased towards twisting- even in the absence of data.
- FIG. 10 illustrates a comparison of minimal volumetric stretching and constant volume constraints as computed by the present invention.
- FIG. 11 illustrates the default LV model recovered using three different volunteer datasets.
- FIGS. 12a and 12b illustrate final fits to volunteer data for the ED and ES phases respectively.
- FIG. 13 illustrates the eigenvalues of the principle components of the strain tensor of the LV at end systole.
- the present invention includes three major elements.
- a novel modeling formulation 10 which includes a global model with parametric offsets.
- This model formulation 10 is capable of describing an expected (or default) configuration which facilitates appropriate model scaling as well as proper model-data correspondences.
- This model formulation 10 may be considered a type of hybrid model in that it is an amalgam of a global (parametric) model and a local (spline-like) model.
- the model formulation 10 is implemented in a solid, thick-walled ellipsoid model for describing the Left Ventricle (LV) of the heart.
- the input to model formulation 10 are MR images of typical LV's.
- This data with parametric offsets 11 and global component 12 provide a scaleable default model 13.
- An input of tagged MR images of the specific LV with scaleable default model 13 and local deformations 14 provide overall model builder 15.
- the output of model formulation 10 is an analysis of LV motion.
- the second major element of the present invention is the introduction of constant volume constraints 17 to cardiac motion recovery. Constant volume constraints 17 interface with overall model builder 15. It is postulated that the volume of a region of tissue remains approximately fixed over the cardiac cycle and this is enforced during motion tracking.
- the third element of the present invention is model tesselator 16 which interfaces with scaleable default model 13.
- Model formulation 10 is described by discrete nodes linearly interpolated to form prismoidal elements. To provide high stability, these elements are arranged in a unique configuration based on the geodesic dome developed by R. Buckminster Fuller as described by H. Kenner in Geodesic Math And How To Use It, Berkeley University of California Press, 1976. Model formulation 10 is demonstrated fitting to segmented tagged-MR image data.
- model structure Under constraints such as constant volume or minimal inter-nodal stretching, the model structure is subject to stress. In order to maintain stability in recovery and lessen the bias inherent in its discrete implementation, it becomes important that the model have sufficient structural support.
- a unique tessellation of the model of the present invention is introduced which provides this support by basing the nodal distribution on geodesic domes.
- the model formulation of the present invention is made up of three components; base global model, parametric offsets, and local deformations. This is illustrated in FIGS. 3a and 3b. From FIG. 3a, the base global model 30 and parametric offsets 32 form the scaleable default model 34. From FIG. 3b, the scaled default model 36 plus local deformations 38 form the overall model 39. The local deformations tailor a scaled default model to a specific dataset. The length and direction of the local deformations are not parametric. More concisely,
- the local deformations are used to tailor the scaled default model to a specific dataset.
- the following will describe each component in detail using the HVV as an example
- the global component of the present invention is augmented with tapering (along the x and y axes), bending and twisting using the variations of these formulations * described by D. Terzopoulos and D. Metaxas.
- the twisting of the inner and outer walls are controlled by independent parameters, twist inner and twist outer , in a fashion similar to Equation 2.
- Parametric offsets (from the global component) are introduced to the standard hybrid model formulation in order to create a default or "rest" shape which resembles the object undergoing recovery. Thus, regions where data is sparse are more likely to be estimated correctly. The inclusion results in a significantly more accurate default shape than could be modeled with the implicit parametric global component alone even with its glob)al deformations (e.g., bending). And, it is possible to describe this complex shape with a very few parameters.
- offsets are described simply by a Cartesian vector (the "offset vector") and a point of attachment to the base global model.
- Parametric offsets are described by a set of intrinsic parameter values with associated points of attachment.
- parametric offsets are a trio (u off v off , ⁇ off ) plus a point of attachment.
- the offset vector itself is calculated by evaluating the base global model (Equation 1) at (u off ,v off , ⁇ off ), and taking the vector difference with the point of attachment (U attach ,v attach , ⁇ attach ). This is illustrated in FIG. 4.
- Calculation of the values (u off ,v off , ⁇ off ) is performed on a dataset describing a typical instance or average of instances of the type of object likely to be recovered. The values are found in the direction of the gradient
- FIGS. 5a, 5b and 5c illustrate the default shape may become extremely distorted as the base global model deforms. This is illustrated in FIGS. 5a, 5b and 5c.
- FIG. 5a illustrates a spherical model with offsets.
- FIG. 5b illustrates a radially scaled model with parametric offsets. Note that the overall structure of the model is preserved.
- FIG. 5c illustrates a radially scaled model with non-parametric offsets. The offsets remain fixed as the global parameters change, resulting in a distorted shape. Note that the scaling demonstrated in this figure could have been implemented as a simple uniform scaling of space. However, global models with parametric offsets are capable of much more sophisticated adjustments.
- FIGS. 6a, 6b and 6c show an example of such an adjustment.
- FIG. 6a illustrates an original model having base global model 60 and offsets 62.
- FIG. 6b illustrates a globally twisted model 64 with parametric offsets 66. The overall structure looks natural since the offsets adjust to the new base component shape.
- FIG. 6c illustrates a globally twisted model 68 with non-parametric offsets 69. The offsets do not adjust to the new global component configuration.
- Local deformations are Cartesian vectors with a point of attachment to the default model. In other formulations they have typically been termed "displacements". This is described by D. Terzopoulos and D. Metaxas and by B. C. Vemuri and A. Radisavljevic. The term "local deformation” is employed in order to clearly distinguish them from offsets.
- Local deformations come into play after the scaling of the default model to a specific dataset.
- the deformations are necessary to tailor the model to a specific dataset if the scaled default shape does not sufficiently approximate the data. Since local deformations cause the model to deviate from the expected shape (the default model), their presence incurs an optional fitting penalty. This is further discussed below.
- Each model node is guaranteed to have no less than two and no more than six elements associated with it. And, the distribution of elements with nodes is guaranteed to be smooth. The result is a structurally sound model.
- the tessellation of the present invention differs from other geodesics in that it is designed for a thick-walled ellipsoid model.
- FIGS. 7a and 7b illustrate a comparison of different model tessellations.
- FIG. 7a illustrates a tessellation based on sealing the end of a tube at a single point. Note that twelve triangular surface elements (underlying prismoidal elements not shown) meet at a single point at the apex 70 as compared to an average of six elsewhere.
- FIG. 7b illustrates a geodesic based tessellation of the model.
- the tessellation of the present invention has significant advantages over the so-called sealed tube approach as described by D. Terzopoulos and D. Metaxas, by J. Park, D. Metaxas and A. Young, by T. McInerney and D. Terzopoulos in "A Finite Element Model For 3d Shape Reconstruction And Nonrigid Motion Tracking", IEEE ICCV, pages 518-523, 1993, and by O'Donnell, Gupta, and Boult in which several model elements may meet at a single node as illustrated in FIG. 7a.
- the sealed tube approach there is no inherent bound on the number of elements associated with the apex node.
- FIG. 8 illustrates a model breaking under high stress during recovery. The breakage initiates at the apex (not shown) of the model where twelve elements meet. Note that the top 82 of the figure is clipped. It is shown below that this situation results in instability and this instability increases with the degree of tessellation. Since a relatively dense tessellation is needed to describe complicated shapes, this limits the effectiveness of the sealed tube approach.
- FIGS. 9a and 9b Another structural bias of the sealed tube approach results in an undesired twisting of the model. This is illustrated in FIGS. 9a and 9b.
- the sealed tube tessellation is biased towards twisting even in the absence of data.
- FIG. 9a illustrates the model prior to the application of the minimal stretching constraint as viewed from the apex.
- FIG. 9b illustrates the model distorting under this constraint.
- the minimal stretching constraint (Equation 3), for example, attempts to maintain initial element edge lengths. In the body of the mesh these constraints are balanced. At the apex, however, there is a resultant bias which causes a differential rotation. (Note that this effect also occurs when using the constant volume constraint).
- the model is tessellated under different schemes for different stages of the recovery process. All of these alternative tessellations are variants of the geodesic approach described above.
- the surfaces of the model are tessellated with planar triangular elements.
- each prismoidal element above is broken down into three tetrahedral subelements. These alternating tessellations (prismoidal, planar triangular surface, tetrahedral) are independent of one another.
- Constraints on deformation are necessary for recovering cardiac motion in a clinical setting.
- a very dense tagged-MR acquisition may take up to two hours. This duration in general is thought to be unacceptable (O. Simonetti PhD. Personal Correspondence, 1995).
- some form of "regularization” must be imposed on the model.
- the minimal stretching constraint as described by O'Donnell, Gupta, and Boult encourages smooth deformations from an initial shape by attempting to maintain inter-nodal distances. It may be used with volumetric as well as planar surface elements. For the present invention, it is employed only on the surface of the model in order to recover a smooth shape. ##STR1## where k s scales the constraint.
- This constraint seeks to minimize the overall change in element volume.
- the element vertices are enforced to deform toward or away from the centroid of the element.
- the constant volume constraint is applied when the difference in volume over time falls above a threshold. Since the cardiac vessels contract and expand over the cycle, the volume of a region of tissue may not be exactly constant but may be assumed to be approximately the same.
- Constant volume forces are especially useful when one or more components of the data motion are unknown, a common characteristic of tagged MR acquisitions. Constant volume constraints allow the model to infer the missing components of motion, something that minimal stretching constraints do much less reliably depending on the element configuration.
- FIG. 10 illustrates a comparison of minimal volumetric stretching and constant volume constraints as computed by the present invention.
- Data forces are applied to pull the bottom corners out of the original shape 80.
- the resulting deformation using minimal stretching constraints is shown for 10 iterations 82 and for 40 iterations 84. Note that the volume increases dramatically.
- the resulting deformation using constant volume constraints is shown for 10 iterations 86 and for 40 iterations 88. As the sides bulge, the element flattens.
- d(u,v,a) is the deformation from the (scaled) default model.
- the final fit may not interpolate the data. If the actual shape of the data deviates from the scaleable default shape, a balance will be struck between the two. To cause the model to favor the default model, a high value of k DisplPenalty may be applied. This implies a strong confidence in the expected shape. Similarly, the final fit can be made to virtually interpolate the data via a low deformation penalty. This is appropriate in the case where the default model has little certainty associated with it as described by S. D. Nova, T. E. Boult and T. O'Donnell in "Physics In A Fantasy World vs. Robust Statistical Estimation", T. Boult, M. Hebert, J. Ponce and A. Gross, editors, 3D Object Representation For Computer Vision, pages 227-296, Springer-Verlag, 1995.
- Constant volume constraints are employed over volumetric minimal stretching because in areas of the model where one or more components of the motion is unknown, a stretching penalty may not influence the shape of the model in a proper way. For example, if the LV model is known to compress in the x direction, it is expected to bulge in the y and z directions to compensate for the absence of any other information. Stretching penalties will not necessarily affect this result whereas constant volume constraints will.
- a default model Prior to the recovery from a specific dataset, a default model must be created. This may be done by fitting the base global model to a set of contour data and allowing the parametric offsets to deform. Since the initial model for these fits is the base global model, it may be necessary to edit the fitting by hand since proper model-data correspondences may not be made.
- Recovery of a specific tagged-MR dataset is composed of two stages. First, the default model is applied to a dataset and allowed to scale. Following this, displacements are used to recover differences between the resulting scaled default model and the data. To estimate the full 3-D motion, it is necessary to simultaneously deform a single model (the HVV) with data from two orthogonal acquisitions. Model deformation at all stages follow the approach developed by D. Terzopoulos and D. Metaxas by minimizing the energy of the model-data system.
- the stability of the tessellation of the present invention was tested by comparing it with the sealed tube tessellation found in O'Donnell, Gupta, and Boult.
- Table 1 shows the results. Breakpoint indicates the number of iterations at which the model collapses (see FIG. 8). The higher the breakpoint number, the more stable the model is to deformation.
- the sealed tube tessellation is described by u and v, the number of nodes in the latitudinal and longitudinal directions respectively.
- the default LV model was recovered by fitting a base global model to a merged set of three segmented LV contour datasets.
- the datasets were from different volunteers and registered by hand using rigid-body rotations as well as scaling. All were from the ED) phase of the cardiac cycle. Two contained short-axis information and one contained long axis information. Some minor editing of the fit was necessary.
- FIG. 11 illustrates the default LV model recovered using 3 different volunteer datasets. The model is in the ED phase of the cardiac cycle.
- the default model is applied to segmented tag intersections as described by G. Funka-Lea and A. Cupta in "The Use Of Hybrid Models To Recover Cardiac Wall Motion In Tagged MR Images", IEEE CVPR, 1996, extracted from long and short axis images. All images were acquired on a Siemens MAGNETOM Vision 1.5 T MRI system with a standard 25 mT/m gradient system. An ECG triggered 2-D gradient echo cine pulse sequence with velocity compensation was utilized. In-plane resolution was 1.74*1.17 mm and slice thickness was 10 mm. The tag grid was applied within 20 msec immediately following the R-wave trigger. The tags were 2 mm wide and spaced 9 mm apart. Six cardiac phases were acquired, covering from ED to ES with 60 msec. temporal resolution. Identical imaging parameters were used for the long-axis and short-axis acquisitions.
- FIGS. 12a and 12b illustrate final fits to volunteer data for the ED (FIG. 12a) and ES (FIG. 12b) phases.
- FIG. 13 is a display of the eigenvalues of the principle components of the strain tensor of the LV at end systole.
- the dark regions represent regions of low strain and the light regions represent regions of high strain.
- the average RMS error of the tag intersection displacements was found to be 0.83 mm and the decrease in volume from ED to ES was 9%.
Abstract
Description
______________________________________ Scaleable = Global Component Default | Model Parametric Offsets Overall Model = Scaleable Default Model + Local Deformations ______________________________________
______________________________________ x(u,v,α) = a.sub.1 (α)cos(u)cos(v) y(u,v,α) = a.sub.2 (α)cos(u)sin(v) z(u,α) = a.sub.3 (α)sin(u) 0 ≦ u ≦ π/2 -π ≦ v ≦ π, (1) ______________________________________
a.sub.i (α)=a.sub.i.sbsb.inner (1-α)+a.sub.i.sbsb.outer α(2)
∂Volume/∂nodes=0 (4)
∫∫∫k.sub.DisplPenalty d(u,v,α)du dv dα(5)
TABLE 1 ______________________________________ A comparison of model stability under the sealed tube and geodesic tessellation schemes. Sealed Tube Thick Geodesic Tessellation Breakpoint Tessellation Breakpoint ______________________________________ u = 4, v = 4 750 17 surf elms 1250 12 elm;1 node 6 elm; 1 node u = 6, v = 8 150 92 surf elms 600 24 elm;1 node 6 elm; 1 node u = 10, v = 15 55 316 surf elms 300 45 elm;1 node 6 elm; 1 node ______________________________________
TABLE 2 ______________________________________ Param Meaning ED ES ______________________________________ al.sub.inner Endocardial rad 2.66 cm 2.58 cm al.sub.outer Epicardial Wall 4.81 cm 4.82 cm a3.sub.inner Apex to base 6.64 cm 6.41 cm taper.sub.x X-axis Tapering -0.21 cm -0.11 cm twist.sub.inner Endocardial twist 0.00 0.016 ______________________________________
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